x minus 8 over x minus 9 is less than 0

Question

algebra equation

Answer 1

(x-8)/(x-9)<0.

If f(x)=(x-8)/(x-9), then the graph of f(x) has two asymptotes: x=9 and y=1.

The x-intercept is at (8,0), so the graph crosses the x-axis at this point and goes from positive to negative. Therefore the inequality is true when x>8; but x is limited by the vertical asymptote x=9. So the solution is:

8<x<9, that is, x∈(8,9) or, in words, x is between 8 and 9 (exclusive).